In this paper, using CalderÃ³n's
reproducing formula and almost orthogonality estimates, we
prove the lifting property and the embedding theorem of the Triebel-Lizorkin
and Besov spaces associated with Zygmund dilations.

A commuting family of subnormal operators need
not have a commuting normal extension. We study when a representation
on an abelian semigroup can be extended to a normal representation,
and show that it suffices to extend the set of generators to
commuting normals. We also extend a result due to Athavale to
representations on abelian lattice ordered semigroups.

In this work we introduce a class of discrete groups containing
subgroups of abstract translations and dilations, respectively. A
variety of wavelet systems can appear as $\pi(\Gamma)\psi$, where $\pi$ is
a unitary representation of a wavelet group and $\Gamma$ is the abstract
pseudo-lattice $\Gamma$. We prove a condition in order that a Parseval
frame $\pi(\Gamma)\psi$ can be dilated to an orthonormal basis of the
form $\tau(\Gamma)\Psi$ where $\tau$ is a super-representation of
$\pi$. For a subclass of groups that includes the case where the
translation subgroup is Heisenberg, we show that this condition
always holds, and we cite familiar examples as applications.